Marvin has some interesting points here that i’ll paste here until i have a few seconds to summarize them.
on cognitive maps:
a more serious problem was that this child had no good image or “cognitive map” of what might result from learning this subject. What function might Math serve in later years? What goals and ambitions might it help to achieve?
and also
More generally, we need to find out more about how each child regards each subject. How might it answer questions like “What am I doing here, and why? ”What can I expect to happen next?” “Where and when am I likely to use this?”
on maths
someone asked if the program was using mathematics. The child replied, “Oh, mathematics isn’t anything special: it’s just the smart way to understand things.”
on the language of math
in mathematics, our children are mentally starved, by having to live in a “linguistic desert.” It really is hard to think about something until one learns enough terms to express the ideas in that this subject.
on getting organezized
If one tries to learn a substantial skill without a good conceptual map, one is likely to end up with several collections of scripts and facts, without good ways to know which of them to use, and when—or how to find good alternatives when what you tried has failed to work.
on special cases
… no small school or community can teach all subjects, or serve the needs of individuals whose abilities are atypical. If a child develops a specialized interest, it is unlikely that any local person can be of much help in developing that child’s special talents and abilities.
on arithmetic
ur usual ways to teach arithmetic only insist on using certain rigid skills, while discouraging each child from trying to invent new ways to do those things.
on worse is better
… one could argue that much of an expert’s competence stems from having learned to avoid the most common bugs. …
on thinking about failure
Many educators have been taught that learning works best when it seems pleasant and enjoyable—but that discounts the value of experiencing frustrations, failures and disappointments.
lots to think about here
